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On the critical point of the Random Walk Pinning Model in dimension d=3
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Quentin Berger, ENS Lyon
Fabio Toninelli, CNRS and ENS Lyon
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Abstract
We consider the Random Walk Pinning Model studied in [Birkner-Sun 2008] and [Birkner-Greven-den Hollander 2008]: this is a random walk X on Z^d, whose law is modified by the exponential of beta times the collision local time up to time N with the (quenched) trajectory Y of another d-dimensional random walk. If beta exceeds a certain critical value beta_c, the two walks stick together for typical Y realizations (localized phase). A natural question is whether the disorder is relevant or not, that is whether the quenched and annealed systems have the same critical behavior. Birkner and Sun proved that beta_c coincides with the critical point of the annealed Random Walk Pinning Model if the space dimension is d=1 or d=2, and that it differs from it in dimension d larger or equal to 4 (for d strictly larger than 4, the result was proven also in [Birkner-Greven-den Hollander 2008]). Here, we consider the open case of the marginal dimension d=3, and we prove non-coincidence of the critical points.
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Full text: PDF
Pages: 654-683
Published on: May 17, 2010
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Electronic Journal of Probability. ISSN: 1083-6489 |
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